Let $G$ denote the Klein's 4-group; that is, let $G \colon= \{e, a, b, ab\} $, $\ a^2 = b^2 = e$, $\ $ $ab = ba$. Then what are all the possible automorphisms of $G$?
My work:
Under an automorphism $T$, the element $a$ has three choices for an image, then $b$ has two, and finally $ab$ has only one, thus giving a total of six possible automorphisms.
Is my reasoning correct?
